Statistical Analysis and Interpretation of Discrete Compositional Data

A composition is a vector of proportions describing the contribution of each of k components to the whole. We introduce an algebra for compositions that provides a natural definition for additive statistical models. The algebra eases interpretation of treatment effects, treatment interactions, and covariates. Our developments extend the logistic normal modeling framework of Aitchison (1982, 1986), and further extend Aitchison’s approach to incorporate discrete observations present in many applications (i.e., counts of objects in different groups). We demonstrate these methods in two examples. The first is a designed experiment evaluating the effect of omnivory on the recovery of arthropod communities to disturbance. The second evaluates the natural variability and spatial dependence of benthic invertebrate communities in the Delaware Bay.